Jmat.Real.erfi, branch x less than or equal to 0.5

Percentage Accurate: 99.8% → 99.9%

Time: 11.2s

Alternatives: 9

Speedup: 4.4×

• 28.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_1\right) + \frac{1}{21} \cdot \left(\left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (* (* t_0 (fabs x)) (fabs x))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+
      (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_0)) (* (/ 1.0 5.0) t_1))
      (* (/ 1.0 21.0) (* (* t_1 (fabs x)) (fabs x))))))))

C

double code(double x) {
	double t_0 = (fabs(x) * fabs(x)) * fabs(x);
	double t_1 = (t_0 * fabs(x)) * fabs(x);
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * fabs(x)) * fabs(x))))));
}

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * Math.abs(x)) * Math.abs(x);
	double t_1 = (t_0 * Math.abs(x)) * Math.abs(x);
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * Math.abs(x)) * Math.abs(x))))));
}

Python

def code(x):
	t_0 = (math.fabs(x) * math.fabs(x)) * math.fabs(x)
	t_1 = (t_0 * math.fabs(x)) * math.fabs(x)
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * math.fabs(x)) * math.fabs(x))))))

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_1 = Float64(Float64(t_0 * abs(x)) * abs(x))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(2.0 / 3.0) * t_0)) + Float64(Float64(1.0 / 5.0) * t_1)) + Float64(Float64(1.0 / 21.0) * Float64(Float64(t_1 * abs(x)) * abs(x))))))
end

MATLAB

function tmp = code(x)
	t_0 = (abs(x) * abs(x)) * abs(x);
	t_1 = (t_0 * abs(x)) * abs(x);
	tmp = abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * abs(x)) * abs(x))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_1\right) + \frac{1}{21} \cdot \left(\left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (* (* t_0 (fabs x)) (fabs x))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+
      (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_0)) (* (/ 1.0 5.0) t_1))
      (* (/ 1.0 21.0) (* (* t_1 (fabs x)) (fabs x))))))))

C

double code(double x) {
	double t_0 = (fabs(x) * fabs(x)) * fabs(x);
	double t_1 = (t_0 * fabs(x)) * fabs(x);
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * fabs(x)) * fabs(x))))));
}

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * Math.abs(x)) * Math.abs(x);
	double t_1 = (t_0 * Math.abs(x)) * Math.abs(x);
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * Math.abs(x)) * Math.abs(x))))));
}

Python

def code(x):
	t_0 = (math.fabs(x) * math.fabs(x)) * math.fabs(x)
	t_1 = (t_0 * math.fabs(x)) * math.fabs(x)
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * math.fabs(x)) * math.fabs(x))))))

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_1 = Float64(Float64(t_0 * abs(x)) * abs(x))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(2.0 / 3.0) * t_0)) + Float64(Float64(1.0 / 5.0) * t_1)) + Float64(Float64(1.0 / 21.0) * Float64(Float64(t_1 * abs(x)) * abs(x))))))
end

MATLAB

function tmp = code(x)
	t_0 = (abs(x) * abs(x)) * abs(x);
	t_1 = (t_0 * abs(x)) * abs(x);
	tmp = abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * abs(x)) * abs(x))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Alternative 1: 99.9% accurate, 4.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \left(x\_m \cdot \left(\left(2 + 0.6666666666666666 \cdot {x\_m}^{2}\right) + \left(0.2 \cdot {x\_m}^{4} + 0.047619047619047616 \cdot {x\_m}^{6}\right)\right)\right) \cdot {\pi}^{-0.5} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (*
  (*
   x_m
   (+
    (+ 2.0 (* 0.6666666666666666 (pow x_m 2.0)))
    (+ (* 0.2 (pow x_m 4.0)) (* 0.047619047619047616 (pow x_m 6.0)))))
  (pow PI -0.5)))

C

x_m = fabs(x);
double code(double x_m) {
	return (x_m * ((2.0 + (0.6666666666666666 * pow(x_m, 2.0))) + ((0.2 * pow(x_m, 4.0)) + (0.047619047619047616 * pow(x_m, 6.0))))) * pow(((double) M_PI), -0.5);
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return (x_m * ((2.0 + (0.6666666666666666 * Math.pow(x_m, 2.0))) + ((0.2 * Math.pow(x_m, 4.0)) + (0.047619047619047616 * Math.pow(x_m, 6.0))))) * Math.pow(Math.PI, -0.5);
}

Python

x_m = math.fabs(x)
def code(x_m):
	return (x_m * ((2.0 + (0.6666666666666666 * math.pow(x_m, 2.0))) + ((0.2 * math.pow(x_m, 4.0)) + (0.047619047619047616 * math.pow(x_m, 6.0))))) * math.pow(math.pi, -0.5)

Julia

x_m = abs(x)
function code(x_m)
	return Float64(Float64(x_m * Float64(Float64(2.0 + Float64(0.6666666666666666 * (x_m ^ 2.0))) + Float64(Float64(0.2 * (x_m ^ 4.0)) + Float64(0.047619047619047616 * (x_m ^ 6.0))))) * (pi ^ -0.5))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (x_m * ((2.0 + (0.6666666666666666 * (x_m ^ 2.0))) + ((0.2 * (x_m ^ 4.0)) + (0.047619047619047616 * (x_m ^ 6.0))))) * (pi ^ -0.5);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[(x$95$m * N[(N[(2.0 + N[(0.6666666666666666 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(0.2 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{\left|x\right|}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|}} \]
4. Add Preprocessing

6. Applied egg-rr 31.5%

   \[\leadsto \color{blue}{\left(x \cdot \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right) \cdot {\pi}^{-0.5}} \]
7. Step-by-step derivation

   1. fma-udef 31.5%

      \[\leadsto \left(x \cdot \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \color{blue}{\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)}\right)\right) \cdot {\pi}^{-0.5} \]

8. Applied egg-rr 31.5%

   \[\leadsto \left(x \cdot \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \color{blue}{\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)}\right)\right) \cdot {\pi}^{-0.5} \]
9. Step-by-step derivation

   1. fma-udef 31.5%

      \[\leadsto \left(x \cdot \left(\color{blue}{\left(0.6666666666666666 \cdot {x}^{2} + 2\right)} + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right) \cdot {\pi}^{-0.5} \]

10. Applied egg-rr 31.5%

    \[\leadsto \left(x \cdot \left(\color{blue}{\left(0.6666666666666666 \cdot {x}^{2} + 2\right)} + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right) \cdot {\pi}^{-0.5} \]

11. Final simplification 31.5%

    \[\leadsto \left(x \cdot \left(\left(2 + 0.6666666666666666 \cdot {x}^{2}\right) + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right) \cdot {\pi}^{-0.5} \]
12. Add Preprocessing

Alternative 2: 98.3% accurate, 4.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := \sqrt{\frac{1}{\pi}}\\ \mathbf{if}\;\left|x\_m\right| \leq 4 \cdot 10^{-8}:\\ \;\;\;\;t\_0 \cdot \left(x\_m \cdot \mathsf{fma}\left(0.6666666666666666, {x\_m}^{2}, 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0 \cdot \left(0.2 \cdot {x\_m}^{5} + 0.047619047619047616 \cdot {x\_m}^{7}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (sqrt (/ 1.0 PI))))
   (if (<= (fabs x_m) 4e-8)
     (* t_0 (* x_m (fma 0.6666666666666666 (pow x_m 2.0) 2.0)))
     (*
      t_0
      (+ (* 0.2 (pow x_m 5.0)) (* 0.047619047619047616 (pow x_m 7.0)))))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = sqrt((1.0 / ((double) M_PI)));
	double tmp;
	if (fabs(x_m) <= 4e-8) {
		tmp = t_0 * (x_m * fma(0.6666666666666666, pow(x_m, 2.0), 2.0));
	} else {
		tmp = t_0 * ((0.2 * pow(x_m, 5.0)) + (0.047619047619047616 * pow(x_m, 7.0)));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	t_0 = sqrt(Float64(1.0 / pi))
	tmp = 0.0
	if (abs(x_m) <= 4e-8)
		tmp = Float64(t_0 * Float64(x_m * fma(0.6666666666666666, (x_m ^ 2.0), 2.0)));
	else
		tmp = Float64(t_0 * Float64(Float64(0.2 * (x_m ^ 5.0)) + Float64(0.047619047619047616 * (x_m ^ 7.0))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 4e-8], N[(t$95$0 * N[(x$95$m * N[(0.6666666666666666 * N[Power[x$95$m, 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[(0.2 * N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x$95$m, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 4.0000000000000001e-8

   1. Initial program 99.9%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.1%

      \[\leadsto \color{blue}{\frac{\left|x\right|}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|}} \]
   4. Add Preprocessing

   6. Applied egg-rr 48.8%

      \[\leadsto \color{blue}{\left(x \cdot \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right) \cdot {\pi}^{-0.5}} \]

   7. Taylor expanded in x around 0 48.8%

      \[\leadsto \color{blue}{0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right) + 2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 48.8%

         \[\leadsto \color{blue}{\left(0.6666666666666666 \cdot {x}^{3}\right) \cdot \sqrt{\frac{1}{\pi}}} + 2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right) \]

      2. associate-*r* 48.8%

         \[\leadsto \left(0.6666666666666666 \cdot {x}^{3}\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}} \]

      3. distribute-rgt-out 48.8%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right)} \]

      4. unpow3 48.8%

         \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} + 2 \cdot x\right) \]

      5. unpow2 48.8%

         \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot \left(\color{blue}{{x}^{2}} \cdot x\right) + 2 \cdot x\right) \]

      6. associate-*r* 48.8%

         \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\left(0.6666666666666666 \cdot {x}^{2}\right) \cdot x} + 2 \cdot x\right) \]

      7. distribute-rgt-in 48.8%

         \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(x \cdot \left(0.6666666666666666 \cdot {x}^{2} + 2\right)\right)} \]

      8. fma-def 48.8%

         \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)}\right) \]

   9. Simplified 48.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)} \]

   if 4.0000000000000001e-8 < (fabs.f64 x)

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{\left|x\right|}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|}} \]
   4. Add Preprocessing

   6. Applied egg-rr 0.1%

      \[\leadsto \color{blue}{\left(x \cdot \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right) \cdot {\pi}^{-0.5}} \]

   7. Taylor expanded in x around inf 0.1%

      \[\leadsto \color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right) + 0.2 \cdot \left({x}^{5} \cdot \sqrt{\frac{1}{\pi}}\right)} \]
   8. Step-by-step derivation

      1. +-commutative 0.1%

         \[\leadsto \color{blue}{0.2 \cdot \left({x}^{5} \cdot \sqrt{\frac{1}{\pi}}\right) + 0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)} \]

      2. associate-*r* 0.1%

         \[\leadsto \color{blue}{\left(0.2 \cdot {x}^{5}\right) \cdot \sqrt{\frac{1}{\pi}}} + 0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right) \]

      3. associate-*r* 0.1%

         \[\leadsto \left(0.2 \cdot {x}^{5}\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}} \]

      4. distribute-rgt-out 0.1%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)} \]

   9. Simplified 0.1%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 98.1% accurate, 4.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x\_m\right| \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(x\_m \cdot \mathsf{fma}\left(0.6666666666666666, {x\_m}^{2}, 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left({x\_m}^{6} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 4e-8)
   (* (sqrt (/ 1.0 PI)) (* x_m (fma 0.6666666666666666 (pow x_m 2.0) 2.0)))
   (* x_m (* (pow x_m 6.0) (/ 0.047619047619047616 (sqrt PI))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 4e-8) {
		tmp = sqrt((1.0 / ((double) M_PI))) * (x_m * fma(0.6666666666666666, pow(x_m, 2.0), 2.0));
	} else {
		tmp = x_m * (pow(x_m, 6.0) * (0.047619047619047616 / sqrt(((double) M_PI))));
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 4e-8)
		tmp = Float64(sqrt(Float64(1.0 / pi)) * Float64(x_m * fma(0.6666666666666666, (x_m ^ 2.0), 2.0)));
	else
		tmp = Float64(x_m * Float64((x_m ^ 6.0) * Float64(0.047619047619047616 / sqrt(pi))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 4e-8], N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(x$95$m * N[(0.6666666666666666 * N[Power[x$95$m, 2.0], $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[Power[x$95$m, 6.0], $MachinePrecision] * N[(0.047619047619047616 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 4.0000000000000001e-8

   1. Initial program 99.9%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.1%

      \[\leadsto \color{blue}{\frac{\left|x\right|}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|}} \]
   4. Add Preprocessing

   6. Applied egg-rr 48.8%

      \[\leadsto \color{blue}{\left(x \cdot \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right) \cdot {\pi}^{-0.5}} \]

   7. Taylor expanded in x around 0 48.8%

      \[\leadsto \color{blue}{0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right) + 2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 48.8%

         \[\leadsto \color{blue}{\left(0.6666666666666666 \cdot {x}^{3}\right) \cdot \sqrt{\frac{1}{\pi}}} + 2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right) \]

      2. associate-*r* 48.8%

         \[\leadsto \left(0.6666666666666666 \cdot {x}^{3}\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}} \]

      3. distribute-rgt-out 48.8%

         \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right)} \]

      4. unpow3 48.8%

         \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} + 2 \cdot x\right) \]

      5. unpow2 48.8%

         \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot \left(\color{blue}{{x}^{2}} \cdot x\right) + 2 \cdot x\right) \]

      6. associate-*r* 48.8%

         \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{\left(0.6666666666666666 \cdot {x}^{2}\right) \cdot x} + 2 \cdot x\right) \]

      7. distribute-rgt-in 48.8%

         \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \color{blue}{\left(x \cdot \left(0.6666666666666666 \cdot {x}^{2} + 2\right)\right)} \]

      8. fma-def 48.8%

         \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \color{blue}{\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)}\right) \]

   9. Simplified 48.8%

      \[\leadsto \color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)} \]

   if 4.0000000000000001e-8 < (fabs.f64 x)

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{\left|x\right|}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 98.2%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)}\right|} \]
   6. Step-by-step derivation

      1. associate-*r* 98.3%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\left(21 \cdot \frac{1}{{x}^{6}}\right) \cdot \sqrt{\pi}}\right|} \]

      2. *-commutative 98.3%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot \left(21 \cdot \frac{1}{{x}^{6}}\right)}\right|} \]

      3. associate-*r/ 98.3%

         \[\leadsto \frac{\left|x\right|}{\left|\sqrt{\pi} \cdot \color{blue}{\frac{21 \cdot 1}{{x}^{6}}}\right|} \]

      4. metadata-eval 98.3%

         \[\leadsto \frac{\left|x\right|}{\left|\sqrt{\pi} \cdot \frac{\color{blue}{21}}{{x}^{6}}\right|} \]

   7. Simplified 98.3%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot \frac{21}{{x}^{6}}}\right|} \]
   8. Step-by-step derivation

      1. associate-*r/ 98.3%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\frac{\sqrt{\pi} \cdot 21}{{x}^{6}}}\right|} \]

   9. Applied egg-rr 98.3%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\frac{\sqrt{\pi} \cdot 21}{{x}^{6}}}\right|} \]

   10. Taylor expanded in x around 0 98.2%

       \[\leadsto \color{blue}{\frac{\left|x\right|}{\left|21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)\right|}} \]
   11. Step-by-step derivation

       1. *-commutative 98.2%

          \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right) \cdot 21}\right|} \]

       2. associate-*l/ 98.2%

          \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\frac{1 \cdot \sqrt{\pi}}{{x}^{6}}} \cdot 21\right|} \]

       3. *-lft-identity 98.2%

          \[\leadsto \frac{\left|x\right|}{\left|\frac{\color{blue}{\sqrt{\pi}}}{{x}^{6}} \cdot 21\right|} \]

       4. associate-/r/ 98.3%

          \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\frac{\sqrt{\pi}}{\frac{{x}^{6}}{21}}}\right|} \]

       5. fabs-div 98.3%

          \[\leadsto \color{blue}{\left|\frac{x}{\frac{\sqrt{\pi}}{\frac{{x}^{6}}{21}}}\right|} \]

       6. rem-square-sqrt 0.0%

          \[\leadsto \left|\color{blue}{\sqrt{\frac{x}{\frac{\sqrt{\pi}}{\frac{{x}^{6}}{21}}}} \cdot \sqrt{\frac{x}{\frac{\sqrt{\pi}}{\frac{{x}^{6}}{21}}}}}\right| \]

       7. fabs-sqr 0.0%

          \[\leadsto \color{blue}{\sqrt{\frac{x}{\frac{\sqrt{\pi}}{\frac{{x}^{6}}{21}}}} \cdot \sqrt{\frac{x}{\frac{\sqrt{\pi}}{\frac{{x}^{6}}{21}}}}} \]

       8. rem-square-sqrt 0.1%

          \[\leadsto \color{blue}{\frac{x}{\frac{\sqrt{\pi}}{\frac{{x}^{6}}{21}}}} \]

       9. *-rgt-identity 0.1%

          \[\leadsto \frac{\color{blue}{x \cdot 1}}{\frac{\sqrt{\pi}}{\frac{{x}^{6}}{21}}} \]

       10. associate-*r/ 0.1%

           \[\leadsto \color{blue}{x \cdot \frac{1}{\frac{\sqrt{\pi}}{\frac{{x}^{6}}{21}}}} \]

       11. associate-/l* 0.1%

           \[\leadsto x \cdot \frac{1}{\color{blue}{\frac{\sqrt{\pi} \cdot 21}{{x}^{6}}}} \]

       12. associate-/r/ 0.1%

           \[\leadsto x \cdot \color{blue}{\left(\frac{1}{\sqrt{\pi} \cdot 21} \cdot {x}^{6}\right)} \]

   12. Simplified 0.1%

       \[\leadsto \color{blue}{x \cdot \left(\frac{0.047619047619047616}{\sqrt{\pi}} \cdot {x}^{6}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right)\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left({x}^{6} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 5.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ {\pi}^{-0.5} \cdot \left(x\_m \cdot \left(2 + \left(0.2 \cdot {x\_m}^{4} + 0.047619047619047616 \cdot {x\_m}^{6}\right)\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (*
  (pow PI -0.5)
  (*
   x_m
   (+ 2.0 (+ (* 0.2 (pow x_m 4.0)) (* 0.047619047619047616 (pow x_m 6.0)))))))

C

x_m = fabs(x);
double code(double x_m) {
	return pow(((double) M_PI), -0.5) * (x_m * (2.0 + ((0.2 * pow(x_m, 4.0)) + (0.047619047619047616 * pow(x_m, 6.0)))));
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.pow(Math.PI, -0.5) * (x_m * (2.0 + ((0.2 * Math.pow(x_m, 4.0)) + (0.047619047619047616 * Math.pow(x_m, 6.0)))));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.pow(math.pi, -0.5) * (x_m * (2.0 + ((0.2 * math.pow(x_m, 4.0)) + (0.047619047619047616 * math.pow(x_m, 6.0)))))

Julia

x_m = abs(x)
function code(x_m)
	return Float64((pi ^ -0.5) * Float64(x_m * Float64(2.0 + Float64(Float64(0.2 * (x_m ^ 4.0)) + Float64(0.047619047619047616 * (x_m ^ 6.0))))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (pi ^ -0.5) * (x_m * (2.0 + ((0.2 * (x_m ^ 4.0)) + (0.047619047619047616 * (x_m ^ 6.0)))));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x$95$m * N[(2.0 + N[(N[(0.2 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{\left|x\right|}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|}} \]
4. Add Preprocessing

6. Applied egg-rr 31.5%

   \[\leadsto \color{blue}{\left(x \cdot \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right) \cdot {\pi}^{-0.5}} \]
7. Step-by-step derivation

   1. fma-udef 31.5%

      \[\leadsto \left(x \cdot \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \color{blue}{\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)}\right)\right) \cdot {\pi}^{-0.5} \]

8. Applied egg-rr 31.5%

   \[\leadsto \left(x \cdot \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \color{blue}{\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)}\right)\right) \cdot {\pi}^{-0.5} \]

9. Taylor expanded in x around 0 31.5%

   \[\leadsto \left(x \cdot \left(\color{blue}{2} + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right) \cdot {\pi}^{-0.5} \]

10. Final simplification 31.5%

    \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \left(2 + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right) \]
11. Add Preprocessing

Alternative 5: 98.1% accurate, 5.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;\left|x\_m\right| \leq 4 \cdot 10^{-8}:\\ \;\;\;\;x\_m \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left({x\_m}^{6} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= (fabs x_m) 4e-8)
   (* x_m (/ 2.0 (sqrt PI)))
   (* x_m (* (pow x_m 6.0) (/ 0.047619047619047616 (sqrt PI))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (fabs(x_m) <= 4e-8) {
		tmp = x_m * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = x_m * (pow(x_m, 6.0) * (0.047619047619047616 / sqrt(((double) M_PI))));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (Math.abs(x_m) <= 4e-8) {
		tmp = x_m * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = x_m * (Math.pow(x_m, 6.0) * (0.047619047619047616 / Math.sqrt(Math.PI)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if math.fabs(x_m) <= 4e-8:
		tmp = x_m * (2.0 / math.sqrt(math.pi))
	else:
		tmp = x_m * (math.pow(x_m, 6.0) * (0.047619047619047616 / math.sqrt(math.pi)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (abs(x_m) <= 4e-8)
		tmp = Float64(x_m * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64(x_m * Float64((x_m ^ 6.0) * Float64(0.047619047619047616 / sqrt(pi))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (abs(x_m) <= 4e-8)
		tmp = x_m * (2.0 / sqrt(pi));
	else
		tmp = x_m * ((x_m ^ 6.0) * (0.047619047619047616 / sqrt(pi)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[N[Abs[x$95$m], $MachinePrecision], 4e-8], N[(x$95$m * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[Power[x$95$m, 6.0], $MachinePrecision] * N[(0.047619047619047616 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 4.0000000000000001e-8

   1. Initial program 99.9%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.1%

      \[\leadsto \color{blue}{\frac{\left|x\right|}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 99.1%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{0.5 \cdot \sqrt{\pi}}\right|} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 99.1%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left|x\right|}{\left|0.5 \cdot \sqrt{\pi}\right|}\right)\right)} \]

      2. expm1-udef 7.6%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left|x\right|}{\left|0.5 \cdot \sqrt{\pi}\right|}\right)} - 1} \]

      3. add-sqr-sqrt 4.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{\left|0.5 \cdot \sqrt{\pi}\right|}\right)} - 1 \]

      4. fabs-sqr 4.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\left|0.5 \cdot \sqrt{\pi}\right|}\right)} - 1 \]

      5. add-sqr-sqrt 7.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{x}}{\left|0.5 \cdot \sqrt{\pi}\right|}\right)} - 1 \]

      6. add-sqr-sqrt 7.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{\left|\color{blue}{\sqrt{0.5 \cdot \sqrt{\pi}} \cdot \sqrt{0.5 \cdot \sqrt{\pi}}}\right|}\right)} - 1 \]

      7. fabs-sqr 7.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{\color{blue}{\sqrt{0.5 \cdot \sqrt{\pi}} \cdot \sqrt{0.5 \cdot \sqrt{\pi}}}}\right)} - 1 \]

      8. add-sqr-sqrt 7.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{\color{blue}{0.5 \cdot \sqrt{\pi}}}\right)} - 1 \]

   7. Applied egg-rr 7.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{0.5 \cdot \sqrt{\pi}}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 48.4%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{0.5 \cdot \sqrt{\pi}}\right)\right)} \]

      2. expm1-log1p 48.4%

         \[\leadsto \color{blue}{\frac{x}{0.5 \cdot \sqrt{\pi}}} \]

      3. *-rgt-identity 48.4%

         \[\leadsto \frac{\color{blue}{x \cdot 1}}{0.5 \cdot \sqrt{\pi}} \]

      4. associate-*r/ 48.8%

         \[\leadsto \color{blue}{x \cdot \frac{1}{0.5 \cdot \sqrt{\pi}}} \]

      5. associate-/r* 48.8%

         \[\leadsto x \cdot \color{blue}{\frac{\frac{1}{0.5}}{\sqrt{\pi}}} \]

      6. metadata-eval 48.8%

         \[\leadsto x \cdot \frac{\color{blue}{2}}{\sqrt{\pi}} \]

   9. Simplified 48.8%

      \[\leadsto \color{blue}{x \cdot \frac{2}{\sqrt{\pi}}} \]

   if 4.0000000000000001e-8 < (fabs.f64 x)

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{\left|x\right|}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 98.2%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)}\right|} \]
   6. Step-by-step derivation

      1. associate-*r* 98.3%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\left(21 \cdot \frac{1}{{x}^{6}}\right) \cdot \sqrt{\pi}}\right|} \]

      2. *-commutative 98.3%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot \left(21 \cdot \frac{1}{{x}^{6}}\right)}\right|} \]

      3. associate-*r/ 98.3%

         \[\leadsto \frac{\left|x\right|}{\left|\sqrt{\pi} \cdot \color{blue}{\frac{21 \cdot 1}{{x}^{6}}}\right|} \]

      4. metadata-eval 98.3%

         \[\leadsto \frac{\left|x\right|}{\left|\sqrt{\pi} \cdot \frac{\color{blue}{21}}{{x}^{6}}\right|} \]

   7. Simplified 98.3%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot \frac{21}{{x}^{6}}}\right|} \]
   8. Step-by-step derivation

      1. associate-*r/ 98.3%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\frac{\sqrt{\pi} \cdot 21}{{x}^{6}}}\right|} \]

   9. Applied egg-rr 98.3%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\frac{\sqrt{\pi} \cdot 21}{{x}^{6}}}\right|} \]

   10. Taylor expanded in x around 0 98.2%

       \[\leadsto \color{blue}{\frac{\left|x\right|}{\left|21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)\right|}} \]
   11. Step-by-step derivation

       1. *-commutative 98.2%

          \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right) \cdot 21}\right|} \]

       2. associate-*l/ 98.2%

          \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\frac{1 \cdot \sqrt{\pi}}{{x}^{6}}} \cdot 21\right|} \]

       3. *-lft-identity 98.2%

          \[\leadsto \frac{\left|x\right|}{\left|\frac{\color{blue}{\sqrt{\pi}}}{{x}^{6}} \cdot 21\right|} \]

       4. associate-/r/ 98.3%

          \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\frac{\sqrt{\pi}}{\frac{{x}^{6}}{21}}}\right|} \]

       5. fabs-div 98.3%

          \[\leadsto \color{blue}{\left|\frac{x}{\frac{\sqrt{\pi}}{\frac{{x}^{6}}{21}}}\right|} \]

       6. rem-square-sqrt 0.0%

          \[\leadsto \left|\color{blue}{\sqrt{\frac{x}{\frac{\sqrt{\pi}}{\frac{{x}^{6}}{21}}}} \cdot \sqrt{\frac{x}{\frac{\sqrt{\pi}}{\frac{{x}^{6}}{21}}}}}\right| \]

       7. fabs-sqr 0.0%

          \[\leadsto \color{blue}{\sqrt{\frac{x}{\frac{\sqrt{\pi}}{\frac{{x}^{6}}{21}}}} \cdot \sqrt{\frac{x}{\frac{\sqrt{\pi}}{\frac{{x}^{6}}{21}}}}} \]

       8. rem-square-sqrt 0.1%

          \[\leadsto \color{blue}{\frac{x}{\frac{\sqrt{\pi}}{\frac{{x}^{6}}{21}}}} \]

       9. *-rgt-identity 0.1%

          \[\leadsto \frac{\color{blue}{x \cdot 1}}{\frac{\sqrt{\pi}}{\frac{{x}^{6}}{21}}} \]

       10. associate-*r/ 0.1%

           \[\leadsto \color{blue}{x \cdot \frac{1}{\frac{\sqrt{\pi}}{\frac{{x}^{6}}{21}}}} \]

       11. associate-/l* 0.1%

           \[\leadsto x \cdot \frac{1}{\color{blue}{\frac{\sqrt{\pi} \cdot 21}{{x}^{6}}}} \]

       12. associate-/r/ 0.1%

           \[\leadsto x \cdot \color{blue}{\left(\frac{1}{\sqrt{\pi} \cdot 21} \cdot {x}^{6}\right)} \]

   12. Simplified 0.1%

       \[\leadsto \color{blue}{x \cdot \left(\frac{0.047619047619047616}{\sqrt{\pi}} \cdot {x}^{6}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 4 \cdot 10^{-8}:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left({x}^{6} \cdot \frac{0.047619047619047616}{\sqrt{\pi}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 96.9% accurate, 8.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.85:\\ \;\;\;\;x\_m \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{x\_m}^{14}}{\pi \cdot 441}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.85)
   (* x_m (/ 2.0 (sqrt PI)))
   (sqrt (/ (pow x_m 14.0) (* PI 441.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.85) {
		tmp = x_m * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = sqrt((pow(x_m, 14.0) / (((double) M_PI) * 441.0)));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.85) {
		tmp = x_m * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = Math.sqrt((Math.pow(x_m, 14.0) / (Math.PI * 441.0)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.85:
		tmp = x_m * (2.0 / math.sqrt(math.pi))
	else:
		tmp = math.sqrt((math.pow(x_m, 14.0) / (math.pi * 441.0)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.85)
		tmp = Float64(x_m * Float64(2.0 / sqrt(pi)));
	else
		tmp = sqrt(Float64((x_m ^ 14.0) / Float64(pi * 441.0)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.85)
		tmp = x_m * (2.0 / sqrt(pi));
	else
		tmp = sqrt(((x_m ^ 14.0) / (pi * 441.0)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.85], N[(x$95$m * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sqrt[N[(N[Power[x$95$m, 14.0], $MachinePrecision] / N[(Pi * 441.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.9%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{\left|x\right|}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 66.0%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{0.5 \cdot \sqrt{\pi}}\right|} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 66.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left|x\right|}{\left|0.5 \cdot \sqrt{\pi}\right|}\right)\right)} \]

      2. expm1-udef 7.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left|x\right|}{\left|0.5 \cdot \sqrt{\pi}\right|}\right)} - 1} \]

      3. add-sqr-sqrt 2.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{\left|0.5 \cdot \sqrt{\pi}\right|}\right)} - 1 \]

      4. fabs-sqr 2.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\left|0.5 \cdot \sqrt{\pi}\right|}\right)} - 1 \]

      5. add-sqr-sqrt 4.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{x}}{\left|0.5 \cdot \sqrt{\pi}\right|}\right)} - 1 \]

      6. add-sqr-sqrt 4.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{\left|\color{blue}{\sqrt{0.5 \cdot \sqrt{\pi}} \cdot \sqrt{0.5 \cdot \sqrt{\pi}}}\right|}\right)} - 1 \]

      7. fabs-sqr 4.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{\color{blue}{\sqrt{0.5 \cdot \sqrt{\pi}} \cdot \sqrt{0.5 \cdot \sqrt{\pi}}}}\right)} - 1 \]

      8. add-sqr-sqrt 4.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{\color{blue}{0.5 \cdot \sqrt{\pi}}}\right)} - 1 \]

   7. Applied egg-rr 4.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{0.5 \cdot \sqrt{\pi}}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 31.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{0.5 \cdot \sqrt{\pi}}\right)\right)} \]

      2. expm1-log1p 31.4%

         \[\leadsto \color{blue}{\frac{x}{0.5 \cdot \sqrt{\pi}}} \]

      3. *-rgt-identity 31.4%

         \[\leadsto \frac{\color{blue}{x \cdot 1}}{0.5 \cdot \sqrt{\pi}} \]

      4. associate-*r/ 31.6%

         \[\leadsto \color{blue}{x \cdot \frac{1}{0.5 \cdot \sqrt{\pi}}} \]

      5. associate-/r* 31.6%

         \[\leadsto x \cdot \color{blue}{\frac{\frac{1}{0.5}}{\sqrt{\pi}}} \]

      6. metadata-eval 31.6%

         \[\leadsto x \cdot \frac{\color{blue}{2}}{\sqrt{\pi}} \]

   9. Simplified 31.6%

      \[\leadsto \color{blue}{x \cdot \frac{2}{\sqrt{\pi}}} \]

   if 1.8500000000000001 < x

   1. Initial program 99.9%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{\left|x\right|}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|}} \]
   4. Add Preprocessing

   6. Applied egg-rr 31.5%

      \[\leadsto \color{blue}{\left(x \cdot \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right) \cdot {\pi}^{-0.5}} \]

   7. Taylor expanded in x around inf 3.7%

      \[\leadsto \color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 3.7%

         \[\leadsto \color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}} \]

   9. Simplified 3.7%

      \[\leadsto \color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 3.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right)} \]

       2. expm1-udef 3.6%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}\right)} - 1} \]

       3. associate-*l* 3.6%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)}\right)} - 1 \]

       4. inv-pow 3.6%

          \[\leadsto e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\color{blue}{{\pi}^{-1}}}\right)\right)} - 1 \]

       5. sqrt-pow1 3.6%

          \[\leadsto e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot \color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}}\right)\right)} - 1 \]

       6. metadata-eval 3.6%

          \[\leadsto e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{\color{blue}{-0.5}}\right)\right)} - 1 \]

   11. Applied egg-rr 3.6%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{-0.5}\right)\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 3.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{-0.5}\right)\right)\right)} \]

       2. expm1-log1p 3.7%

          \[\leadsto \color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{-0.5}\right)} \]

       3. associate-*r* 3.7%

          \[\leadsto \color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot {\pi}^{-0.5}} \]

   13. Simplified 3.7%

       \[\leadsto \color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot {\pi}^{-0.5}} \]

   15. Applied egg-rr 35.8%

       \[\leadsto \color{blue}{\sqrt{\frac{\frac{{x}^{14}}{441}}{\pi}}} \]
   16. Step-by-step derivation

       1. associate-/l/ 35.8%

          \[\leadsto \sqrt{\color{blue}{\frac{{x}^{14}}{\pi \cdot 441}}} \]

   17. Simplified 35.8%

       \[\leadsto \color{blue}{\sqrt{\frac{{x}^{14}}{\pi \cdot 441}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{{x}^{14}}{\pi \cdot 441}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 98.9% accurate, 8.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.85:\\ \;\;\;\;x\_m \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \frac{{x\_m}^{7}}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.85)
   (* x_m (/ 2.0 (sqrt PI)))
   (* 0.047619047619047616 (/ (pow x_m 7.0) (sqrt PI)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.85) {
		tmp = x_m * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = 0.047619047619047616 * (pow(x_m, 7.0) / sqrt(((double) M_PI)));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.85) {
		tmp = x_m * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = 0.047619047619047616 * (Math.pow(x_m, 7.0) / Math.sqrt(Math.PI));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.85:
		tmp = x_m * (2.0 / math.sqrt(math.pi))
	else:
		tmp = 0.047619047619047616 * (math.pow(x_m, 7.0) / math.sqrt(math.pi))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.85)
		tmp = Float64(x_m * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64(0.047619047619047616 * Float64((x_m ^ 7.0) / sqrt(pi)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.85)
		tmp = x_m * (2.0 / sqrt(pi));
	else
		tmp = 0.047619047619047616 * ((x_m ^ 7.0) / sqrt(pi));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.85], N[(x$95$m * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.047619047619047616 * N[(N[Power[x$95$m, 7.0], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.9%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{\left|x\right|}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 66.0%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{0.5 \cdot \sqrt{\pi}}\right|} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 66.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left|x\right|}{\left|0.5 \cdot \sqrt{\pi}\right|}\right)\right)} \]

      2. expm1-udef 7.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left|x\right|}{\left|0.5 \cdot \sqrt{\pi}\right|}\right)} - 1} \]

      3. add-sqr-sqrt 2.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{\left|0.5 \cdot \sqrt{\pi}\right|}\right)} - 1 \]

      4. fabs-sqr 2.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\left|0.5 \cdot \sqrt{\pi}\right|}\right)} - 1 \]

      5. add-sqr-sqrt 4.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{x}}{\left|0.5 \cdot \sqrt{\pi}\right|}\right)} - 1 \]

      6. add-sqr-sqrt 4.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{\left|\color{blue}{\sqrt{0.5 \cdot \sqrt{\pi}} \cdot \sqrt{0.5 \cdot \sqrt{\pi}}}\right|}\right)} - 1 \]

      7. fabs-sqr 4.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{\color{blue}{\sqrt{0.5 \cdot \sqrt{\pi}} \cdot \sqrt{0.5 \cdot \sqrt{\pi}}}}\right)} - 1 \]

      8. add-sqr-sqrt 4.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{\color{blue}{0.5 \cdot \sqrt{\pi}}}\right)} - 1 \]

   7. Applied egg-rr 4.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{0.5 \cdot \sqrt{\pi}}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 31.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{0.5 \cdot \sqrt{\pi}}\right)\right)} \]

      2. expm1-log1p 31.4%

         \[\leadsto \color{blue}{\frac{x}{0.5 \cdot \sqrt{\pi}}} \]

      3. *-rgt-identity 31.4%

         \[\leadsto \frac{\color{blue}{x \cdot 1}}{0.5 \cdot \sqrt{\pi}} \]

      4. associate-*r/ 31.6%

         \[\leadsto \color{blue}{x \cdot \frac{1}{0.5 \cdot \sqrt{\pi}}} \]

      5. associate-/r* 31.6%

         \[\leadsto x \cdot \color{blue}{\frac{\frac{1}{0.5}}{\sqrt{\pi}}} \]

      6. metadata-eval 31.6%

         \[\leadsto x \cdot \frac{\color{blue}{2}}{\sqrt{\pi}} \]

   9. Simplified 31.6%

      \[\leadsto \color{blue}{x \cdot \frac{2}{\sqrt{\pi}}} \]

   if 1.8500000000000001 < x

   1. Initial program 99.9%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{\left|x\right|}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|}} \]
   4. Add Preprocessing

   6. Applied egg-rr 31.5%

      \[\leadsto \color{blue}{\left(x \cdot \left(\mathsf{fma}\left(0.6666666666666666, {x}^{2}, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right) \cdot {\pi}^{-0.5}} \]

   7. Taylor expanded in x around inf 3.7%

      \[\leadsto \color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)} \]
   8. Step-by-step derivation

      1. associate-*r* 3.7%

         \[\leadsto \color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}} \]

   9. Simplified 3.7%

      \[\leadsto \color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}} \]
   10. Step-by-step derivation

       1. expm1-log1p-u 3.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right)} \]

       2. expm1-udef 3.6%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}\right)} - 1} \]

       3. associate-*l* 3.6%

          \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)}\right)} - 1 \]

       4. inv-pow 3.6%

          \[\leadsto e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\color{blue}{{\pi}^{-1}}}\right)\right)} - 1 \]

       5. sqrt-pow1 3.6%

          \[\leadsto e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot \color{blue}{{\pi}^{\left(\frac{-1}{2}\right)}}\right)\right)} - 1 \]

       6. metadata-eval 3.6%

          \[\leadsto e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{\color{blue}{-0.5}}\right)\right)} - 1 \]

   11. Applied egg-rr 3.6%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{-0.5}\right)\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 3.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{-0.5}\right)\right)\right)} \]

       2. expm1-log1p 3.7%

          \[\leadsto \color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{-0.5}\right)} \]

       3. associate-*r* 3.7%

          \[\leadsto \color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot {\pi}^{-0.5}} \]

   13. Simplified 3.7%

       \[\leadsto \color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot {\pi}^{-0.5}} \]

   15. Applied egg-rr 3.6%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{{x}^{7}}{\sqrt{\pi} \cdot 21}\right)} - 1} \]
   16. Step-by-step derivation

       1. expm1-def 3.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{{x}^{7}}{\sqrt{\pi} \cdot 21}\right)\right)} \]

       2. expm1-log1p 3.7%

          \[\leadsto \color{blue}{\frac{{x}^{7}}{\sqrt{\pi} \cdot 21}} \]

       3. *-lft-identity 3.7%

          \[\leadsto \frac{\color{blue}{1 \cdot {x}^{7}}}{\sqrt{\pi} \cdot 21} \]

       4. *-commutative 3.7%

          \[\leadsto \frac{1 \cdot {x}^{7}}{\color{blue}{21 \cdot \sqrt{\pi}}} \]

       5. times-frac 3.7%

          \[\leadsto \color{blue}{\frac{1}{21} \cdot \frac{{x}^{7}}{\sqrt{\pi}}} \]

       6. metadata-eval 3.7%

          \[\leadsto \color{blue}{0.047619047619047616} \cdot \frac{{x}^{7}}{\sqrt{\pi}} \]

   17. Simplified 3.7%

       \[\leadsto \color{blue}{0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 98.9% accurate, 8.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.85:\\ \;\;\;\;x\_m \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.047619047619047616 \cdot {x\_m}^{7}}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.85)
   (* x_m (/ 2.0 (sqrt PI)))
   (/ (* 0.047619047619047616 (pow x_m 7.0)) (sqrt PI))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.85) {
		tmp = x_m * (2.0 / sqrt(((double) M_PI)));
	} else {
		tmp = (0.047619047619047616 * pow(x_m, 7.0)) / sqrt(((double) M_PI));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.85) {
		tmp = x_m * (2.0 / Math.sqrt(Math.PI));
	} else {
		tmp = (0.047619047619047616 * Math.pow(x_m, 7.0)) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.85:
		tmp = x_m * (2.0 / math.sqrt(math.pi))
	else:
		tmp = (0.047619047619047616 * math.pow(x_m, 7.0)) / math.sqrt(math.pi)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.85)
		tmp = Float64(x_m * Float64(2.0 / sqrt(pi)));
	else
		tmp = Float64(Float64(0.047619047619047616 * (x_m ^ 7.0)) / sqrt(pi));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.85)
		tmp = x_m * (2.0 / sqrt(pi));
	else
		tmp = (0.047619047619047616 * (x_m ^ 7.0)) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.85], N[(x$95$m * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.047619047619047616 * N[Power[x$95$m, 7.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.9%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{\left|x\right|}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 66.0%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{0.5 \cdot \sqrt{\pi}}\right|} \]
   6. Step-by-step derivation

      1. expm1-log1p-u 66.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left|x\right|}{\left|0.5 \cdot \sqrt{\pi}\right|}\right)\right)} \]

      2. expm1-udef 7.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left|x\right|}{\left|0.5 \cdot \sqrt{\pi}\right|}\right)} - 1} \]

      3. add-sqr-sqrt 2.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{\left|0.5 \cdot \sqrt{\pi}\right|}\right)} - 1 \]

      4. fabs-sqr 2.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\left|0.5 \cdot \sqrt{\pi}\right|}\right)} - 1 \]

      5. add-sqr-sqrt 4.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{x}}{\left|0.5 \cdot \sqrt{\pi}\right|}\right)} - 1 \]

      6. add-sqr-sqrt 4.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{\left|\color{blue}{\sqrt{0.5 \cdot \sqrt{\pi}} \cdot \sqrt{0.5 \cdot \sqrt{\pi}}}\right|}\right)} - 1 \]

      7. fabs-sqr 4.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{\color{blue}{\sqrt{0.5 \cdot \sqrt{\pi}} \cdot \sqrt{0.5 \cdot \sqrt{\pi}}}}\right)} - 1 \]

      8. add-sqr-sqrt 4.5%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{\color{blue}{0.5 \cdot \sqrt{\pi}}}\right)} - 1 \]

   7. Applied egg-rr 4.5%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{0.5 \cdot \sqrt{\pi}}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 31.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{0.5 \cdot \sqrt{\pi}}\right)\right)} \]

      2. expm1-log1p 31.4%

         \[\leadsto \color{blue}{\frac{x}{0.5 \cdot \sqrt{\pi}}} \]

      3. *-rgt-identity 31.4%

         \[\leadsto \frac{\color{blue}{x \cdot 1}}{0.5 \cdot \sqrt{\pi}} \]

      4. associate-*r/ 31.6%

         \[\leadsto \color{blue}{x \cdot \frac{1}{0.5 \cdot \sqrt{\pi}}} \]

      5. associate-/r* 31.6%

         \[\leadsto x \cdot \color{blue}{\frac{\frac{1}{0.5}}{\sqrt{\pi}}} \]

      6. metadata-eval 31.6%

         \[\leadsto x \cdot \frac{\color{blue}{2}}{\sqrt{\pi}} \]

   9. Simplified 31.6%

      \[\leadsto \color{blue}{x \cdot \frac{2}{\sqrt{\pi}}} \]

   if 1.8500000000000001 < x

   1. Initial program 99.9%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Simplified 99.4%

      \[\leadsto \color{blue}{\frac{\left|x\right|}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 38.7%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{21 \cdot \left(\frac{1}{{x}^{6}} \cdot \sqrt{\pi}\right)}\right|} \]
   6. Step-by-step derivation

      1. associate-*r* 38.7%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\left(21 \cdot \frac{1}{{x}^{6}}\right) \cdot \sqrt{\pi}}\right|} \]

      2. *-commutative 38.7%

         \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot \left(21 \cdot \frac{1}{{x}^{6}}\right)}\right|} \]

      3. associate-*r/ 38.7%

         \[\leadsto \frac{\left|x\right|}{\left|\sqrt{\pi} \cdot \color{blue}{\frac{21 \cdot 1}{{x}^{6}}}\right|} \]

      4. metadata-eval 38.7%

         \[\leadsto \frac{\left|x\right|}{\left|\sqrt{\pi} \cdot \frac{\color{blue}{21}}{{x}^{6}}\right|} \]

   7. Simplified 38.7%

      \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{\sqrt{\pi} \cdot \frac{21}{{x}^{6}}}\right|} \]
   8. Step-by-step derivation

      1. expm1-log1p-u 38.3%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left|x\right|}{\left|\sqrt{\pi} \cdot \frac{21}{{x}^{6}}\right|}\right)\right)} \]

      2. expm1-udef 38.1%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left|x\right|}{\left|\sqrt{\pi} \cdot \frac{21}{{x}^{6}}\right|}\right)} - 1} \]

      3. add-sqr-sqrt 1.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{\left|\sqrt{\pi} \cdot \frac{21}{{x}^{6}}\right|}\right)} - 1 \]

      4. fabs-sqr 1.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\left|\sqrt{\pi} \cdot \frac{21}{{x}^{6}}\right|}\right)} - 1 \]

      5. add-sqr-sqrt 3.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{x}}{\left|\sqrt{\pi} \cdot \frac{21}{{x}^{6}}\right|}\right)} - 1 \]

      6. add-sqr-sqrt 3.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{\left|\color{blue}{\sqrt{\sqrt{\pi} \cdot \frac{21}{{x}^{6}}} \cdot \sqrt{\sqrt{\pi} \cdot \frac{21}{{x}^{6}}}}\right|}\right)} - 1 \]

      7. fabs-sqr 3.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{\color{blue}{\sqrt{\sqrt{\pi} \cdot \frac{21}{{x}^{6}}} \cdot \sqrt{\sqrt{\pi} \cdot \frac{21}{{x}^{6}}}}}\right)} - 1 \]

      8. add-sqr-sqrt 3.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{\color{blue}{\sqrt{\pi} \cdot \frac{21}{{x}^{6}}}}\right)} - 1 \]

      9. div-inv 3.6%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{\sqrt{\pi} \cdot \color{blue}{\left(21 \cdot \frac{1}{{x}^{6}}\right)}}\right)} - 1 \]

      10. pow-flip 3.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{\sqrt{\pi} \cdot \left(21 \cdot \color{blue}{{x}^{\left(-6\right)}}\right)}\right)} - 1 \]

      11. metadata-eval 3.6%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{\sqrt{\pi} \cdot \left(21 \cdot {x}^{\color{blue}{-6}}\right)}\right)} - 1 \]

   9. Applied egg-rr 3.6%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{\sqrt{\pi} \cdot \left(21 \cdot {x}^{-6}\right)}\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 3.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\sqrt{\pi} \cdot \left(21 \cdot {x}^{-6}\right)}\right)\right)} \]

       2. expm1-log1p 3.7%

          \[\leadsto \color{blue}{\frac{x}{\sqrt{\pi} \cdot \left(21 \cdot {x}^{-6}\right)}} \]

       3. *-commutative 3.7%

          \[\leadsto \frac{x}{\color{blue}{\left(21 \cdot {x}^{-6}\right) \cdot \sqrt{\pi}}} \]

       4. associate-/r* 3.7%

          \[\leadsto \color{blue}{\frac{\frac{x}{21 \cdot {x}^{-6}}}{\sqrt{\pi}}} \]

       5. *-commutative 3.7%

          \[\leadsto \frac{\frac{x}{\color{blue}{{x}^{-6} \cdot 21}}}{\sqrt{\pi}} \]

       6. associate-/r* 3.7%

          \[\leadsto \frac{\color{blue}{\frac{\frac{x}{{x}^{-6}}}{21}}}{\sqrt{\pi}} \]

   11. Simplified 3.7%

       \[\leadsto \color{blue}{\frac{\frac{\frac{x}{{x}^{-6}}}{21}}{\sqrt{\pi}}} \]
   12. Step-by-step derivation

       1. div-inv 3.7%

          \[\leadsto \frac{\color{blue}{\frac{x}{{x}^{-6}} \cdot \frac{1}{21}}}{\sqrt{\pi}} \]

       2. metadata-eval 3.7%

          \[\leadsto \frac{\frac{x}{{x}^{-6}} \cdot \color{blue}{0.047619047619047616}}{\sqrt{\pi}} \]

       3. pow1 3.7%

          \[\leadsto \frac{\frac{\color{blue}{{x}^{1}}}{{x}^{-6}} \cdot 0.047619047619047616}{\sqrt{\pi}} \]

       4. pow-div 3.7%

          \[\leadsto \frac{\color{blue}{{x}^{\left(1 - -6\right)}} \cdot 0.047619047619047616}{\sqrt{\pi}} \]

       5. metadata-eval 3.7%

          \[\leadsto \frac{{x}^{\color{blue}{7}} \cdot 0.047619047619047616}{\sqrt{\pi}} \]

   13. Applied egg-rr 3.7%

       \[\leadsto \frac{\color{blue}{{x}^{7} \cdot 0.047619047619047616}}{\sqrt{\pi}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 31.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;x \cdot \frac{2}{\sqrt{\pi}}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 67.3% accurate, 17.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x\_m \cdot \frac{2}{\sqrt{\pi}} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* x_m (/ 2.0 (sqrt PI))))

C

x_m = fabs(x);
double code(double x_m) {
	return x_m * (2.0 / sqrt(((double) M_PI)));
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return x_m * (2.0 / Math.sqrt(Math.PI));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return x_m * (2.0 / math.sqrt(math.pi))

Julia

x_m = abs(x)
function code(x_m)
	return Float64(x_m * Float64(2.0 / sqrt(pi)))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = x_m * (2.0 / sqrt(pi));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(x$95$m * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Simplified 99.4%

   \[\leadsto \color{blue}{\frac{\left|x\right|}{\left|\frac{\sqrt{\pi}}{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}\right|}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 66.0%

   \[\leadsto \frac{\left|x\right|}{\left|\color{blue}{0.5 \cdot \sqrt{\pi}}\right|} \]
6. Step-by-step derivation

   1. expm1-log1p-u 66.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left|x\right|}{\left|0.5 \cdot \sqrt{\pi}\right|}\right)\right)} \]

   2. expm1-udef 7.1%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left|x\right|}{\left|0.5 \cdot \sqrt{\pi}\right|}\right)} - 1} \]

   3. add-sqr-sqrt 2.5%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right|}{\left|0.5 \cdot \sqrt{\pi}\right|}\right)} - 1 \]

   4. fabs-sqr 2.5%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{\left|0.5 \cdot \sqrt{\pi}\right|}\right)} - 1 \]

   5. add-sqr-sqrt 4.5%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{x}}{\left|0.5 \cdot \sqrt{\pi}\right|}\right)} - 1 \]

   6. add-sqr-sqrt 4.5%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{\left|\color{blue}{\sqrt{0.5 \cdot \sqrt{\pi}} \cdot \sqrt{0.5 \cdot \sqrt{\pi}}}\right|}\right)} - 1 \]

   7. fabs-sqr 4.5%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{\color{blue}{\sqrt{0.5 \cdot \sqrt{\pi}} \cdot \sqrt{0.5 \cdot \sqrt{\pi}}}}\right)} - 1 \]

   8. add-sqr-sqrt 4.5%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{x}{\color{blue}{0.5 \cdot \sqrt{\pi}}}\right)} - 1 \]

7. Applied egg-rr 4.5%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{x}{0.5 \cdot \sqrt{\pi}}\right)} - 1} \]
8. Step-by-step derivation

   1. expm1-def 31.2%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{0.5 \cdot \sqrt{\pi}}\right)\right)} \]

   2. expm1-log1p 31.4%

      \[\leadsto \color{blue}{\frac{x}{0.5 \cdot \sqrt{\pi}}} \]

   3. *-rgt-identity 31.4%

      \[\leadsto \frac{\color{blue}{x \cdot 1}}{0.5 \cdot \sqrt{\pi}} \]

   4. associate-*r/ 31.6%

      \[\leadsto \color{blue}{x \cdot \frac{1}{0.5 \cdot \sqrt{\pi}}} \]

   5. associate-/r* 31.6%

      \[\leadsto x \cdot \color{blue}{\frac{\frac{1}{0.5}}{\sqrt{\pi}}} \]

   6. metadata-eval 31.6%

      \[\leadsto x \cdot \frac{\color{blue}{2}}{\sqrt{\pi}} \]

9. Simplified 31.6%

   \[\leadsto \color{blue}{x \cdot \frac{2}{\sqrt{\pi}}} \]

10. Final simplification 31.6%

    \[\leadsto x \cdot \frac{2}{\sqrt{\pi}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024027 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x less than or equal to 0.5"
  :precision binary64
  :pre (<= x 0.5)
  (fabs (* (/ 1.0 (sqrt PI)) (+ (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) (* (* (fabs x) (fabs x)) (fabs x)))) (* (/ 1.0 5.0) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))) (* (/ 1.0 21.0) (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)))))))